1 / 75

CHAPTER 2 MATRICES

CHAPTER 2 MATRICES. 2.1 Operations with Matrices 2.2 Properties of Matrix Operations 2.3 The Inverse of a Matrix 2.4 Elementary Matrices 2.5 Applications of Matrix Operations.

Download Presentation

CHAPTER 2 MATRICES

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. CHAPTER 2MATRICES 2.1 Operations with Matrices 2.2 Properties of Matrix Operations 2.3 The Inverse of a Matrix 2.4 Elementary Matrices 2.5Applications of Matrix Operations Elementary Linear Algebra 投影片設計編製者 R. Larson (7 Edition) 淡江大學 電機系 翁慶昌 教授

  2. CH 2 Linear Algebra Applied Flight Crew Scheduling (p.47) Beam Deflection (p.64) Information Retrieval (p.58) Computational Fluid Dynamics (p.79) Data Encryption (p.87)

  3. 2.1 Operations with Matrices (i, j)-th entry: • Matrix: row: m column: n size: m×n Elementary Linear Algebra: Section 2.1, p.40

  4. j-th column vector • i-throw vector row matrix column matrix • Square matrix:m = n Elementary Linear Algebra: Section 2.1, p.40

  5. Diagonal matrix: • Trace: Elementary Linear Algebra: Section 2.1, Addition

  6. Ex: Elementary Linear Algebra: Section 2.1, Addition

  7. Equal matrix: • Ex 1: (Equal matrix) Elementary Linear Algebra: Section 2.1, p.40

  8. Matrix addition: • Ex 2: (Matrix addition) Elementary Linear Algebra: Section 2.1, p.41

  9. Matrix subtraction: • Ex 3: (Scalar multiplication and matrix subtraction) Find (a) 3A, (b) –B, (c) 3A–B • Scalar multiplication: Elementary Linear Algebra: Section 2.1, p.41

  10. Sol: (a) (b) (c) Elementary Linear Algebra: Section 2.1, p.41

  11. Size of AB • Notes: (1) A+B = B+A, (2) • Matrix multiplication: where Elementary Linear Algebra: Section 2.1, p.42 & p.44

  12. Ex 4: (Find AB) Sol: Elementary Linear Algebra: Section 2.1, p.43

  13. = = = x A b • Matrix form of a system of linear equations: Elementary Linear Algebra: Section 2.1, p.45

  14. Partitioned matrices: submatrix Elementary Linear Algebra: Section 2.1, Addition

  15. Linear combination of column vectors: Elementary Linear Algebra: Section 2.1, p.46

  16. Ex 7: (Solve a system of linear equations) (infinitely many solutions) Elementary Linear Algebra: Section 2.1, p.47

  17. Key Learning in Section 2.1 • Determine whether two matrices are equal. • Add and subtract matrices and multiply a matrix by a scalar. • Multiply two matrices. • Use matrices to solve a system of linear equations. • Partition a matrix and write a linear combination of column vectors..

  18. Keywords in Section 2.1 row vector: 列向量 column vector: 行向量 diagonal matrix: 對角矩陣 trace: 跡數 equality of matrices: 相等矩陣 matrix addition: 矩陣相加 scalar multiplication: 純量乘法(純量積) matrix subtraction: 矩陣相減 matrix multiplication: 矩陣相乘 partitionedmatrix: 分割矩陣 linear combination: 線性組合

  19. 2.2 Properties of Matrix Operations • Three basic matrix operators: (1) matrix addition (2) scalar multiplication (3) matrix multiplication • Zero matrix: • Identity matrix of order n: Elementary Linear Algebra: Section 2.2, 52-55

  20. Properties of matrix addition and scalar multiplication: Then (1) A+B = B + A (2) A + ( B + C ) = ( A + B ) + C (3) ( cd ) A = c ( dA ) (4) 1A = A (5) c( A+B ) = cA + cB (6) ( c+d ) A = cA + dA Elementary Linear Algebra: Section 2.2, p.52

  21. Notes: • 0m×n: the additive identity for the set of all m×n matrices • –A: the additive inverse of A • Properties of zero matrices: Elementary Linear Algebra: Section 2.2, p.53

  22. (1) A(BC) = (AB)C (2) A(B+C) = AB + AC (3) (A+B)C = AC + BC (4) c (AB) = (cA) B = A (cB) • Properties of matrix multiplication: • Properties of identity matrix: Elementary Linear Algebra: Section 2.2, p.54 & p.56

  23. Transpose of a matrix: Elementary Linear Algebra: Section 2.2, p.57

  24. Sol: (a) (b) (c) • Ex 8: (Find the transpose of the following matrix) (a) (b) (c) Elementary Linear Algebra: Section 2.2, p.57

  25. Properties of transposes: Elementary Linear Algebra: Section 2.2, p.57

  26. Ex: is symmetric, finda, b, c? Sol: • Symmetric matrix: A square matrix A is symmetric if A = AT • Skew-symmetric matrix: A square matrix A is skew-symmetric if AT = –A Elementary Linear Algebra: Section 2.2, Addition

  27. Note: is symmetric Pf: • Ex: is a skew-symmetric, finda, b, c? Sol: Elementary Linear Algebra: Section 2.2, Addition

  28. Matrix: Three situations: • Real number: ab = ba (Commutative law for multiplication) (Sizes are not the same) (Sizes are the same, but matrices are not equal) Elementary Linear Algebra: Section 2.2, Addition

  29. Note: • Ex 4: Sow thatAB andBA are not equal for the matrices. and Sol: Elementary Linear Algebra: Section 2.2, p.55

  30. Real number: (Cancellation law) • Matrix: (1) If C is invertible, then A = B (Cancellation is not valid) Elementary Linear Algebra: Section 2.2, p.55

  31. Sol: So But • Ex 5:(An example in which cancellation is not valid) Show that AC=BC Elementary Linear Algebra: Section 2.2, p.55

  32. Key Learning in Section 2.2 • Use the properties of matrix addition, scalar multiplication, and zero matrices. • Use the properties of matrix multiplication and the identity matrix. • Find the transpose of a matrix.

  33. Keywords in Section 2.2 zero matrix: 零矩陣 identity matrix: 單位矩陣 transpose matrix: 轉置矩陣 symmetric matrix: 對稱矩陣 skew-symmetric matrix: 反對稱矩陣

  34. 2.3 The Inverse of a Matrix • Note: A matrix that does not have an inverse is called noninvertible (or singular). • Inverse matrix: Consider Then (1) A is invertible (or nonsingular) (2) B is the inverse of A Elementary Linear Algebra: Section 2.3, p.62

  35. Pf: • Notes: (1) The inverse of A is denoted by • Thm 2.7: (The inverse of a matrix is unique) If B andC are both inverses of the matrix A, thenB = C. Consequently, the inverse of a matrix is unique. Elementary Linear Algebra: Section 2.3, pp.62-63

  36. Sol: • Find the inverse of a matrix by Gauss-Jordan Elimination: • Ex 2: (Find the inverse of the matrix) Elementary Linear Algebra: Section 2.3, pp.63-64

  37. Thus Elementary Linear Algebra: Section 2.3, pp.63-64

  38. Note: If A can’t be row reduced to I, then A is singular. Elementary Linear Algebra: Section 2.3, p.64

  39. Sol: • Ex 3: (Find the inverse of the following matrix) Elementary Linear Algebra: Section 2.3, p.65

  40. Check: So the matrix A is invertible, and its inverse is Elementary Linear Algebra: Section 2.3, p.65

  41. Power of a square matrix: Elementary Linear Algebra: Section 2.3, Addition

  42. Thm 2.8:(Properties of inverse matrices) If A is an invertible matrix, k is a positive integer, and c is a scalar not equal to zero, then Elementary Linear Algebra: Section 2.3, p.67

  43. Pf: • Note: • Thm 2.9: (The inverse of a product) If A andB are invertible matrices of size n, thenAB is invertible and Elementary Linear Algebra: Section 2.3, p.68

  44. Thm 2.10 (Cancellation properties) If C is an invertible matrix, then the following properties hold: (1) If AC=BC, then A=B (Right cancellation property) (2) If CA=CB, then A=B (Left cancellation property) Pf: • Note: IfC is not invertible, then cancellation is not valid. Elementary Linear Algebra: Section 2.3, p.69

  45. Pf: ( A is nonsingular) • Thm 2.11: (Systems of equations with unique solutions) IfA is an invertible matrix, then the system of linear equations Ax = b has a unique solution given by (Left cancellation property) This solution is unique. Elementary Linear Algebra: Section 2.3, p.70

  46. Note: For square systems (those having the same number of equations as variables), Theorem 2.11 can be used to determine whether the system has a unique solution. • Note: (A is an invertible matrix) Elementary Linear Algebra: Section 2.3, p.70

  47. Key Learning in Section 2.3 • Find the inverse of a matrix (if it exists). • Use properties of inverse matrices. • Use an inverse matrix to solve a system of linear equations.

  48. Keywords in Section 2.3 inverse matrix: 反矩陣 invertible: 可逆 nonsingular: 非奇異 noninvertible: 不可逆 singular: 奇異 power: 冪次

  49. Three row elementary matrices: Interchange two rows. Multiply a row by a nonzero constant. Add a multiple of a row to another row. 2.4 Elementary Matrices • Row elementary matrix: An nnmatrix is called an elementary matrix if it can be obtained from the identity matrixIn by a single elementary operation. • Note: Only do a single elementary row operation. Elementary Linear Algebra: Section 2.4, p.74

  50. Ex 1: (Elementary matrices and nonelementary matrices) Elementary Linear Algebra: Section 2.4, p.74

More Related